Question about the assumptions in a linear regression model 
Above is a screenshot from Introduction to Linear Regression by Douglas C. We usually assume that in linear regression model $y$ has a linear relationship with the parameter, but why here the author says $y$ should have a linear relationship with the regressor $x$?
 A: Linear models make the assumption of a linear relationship because linear models estimate functions of the form $y = a + b_1 x_1 + b_2 x_2 + \cdots +b_n x_n + \text{error}$, i.e. linear functions. If you don't have a linear relationship, you need to use a different model or somehow transform $y$ and/or $x$ so that the model is linear.
A: I think you are possibly confusing two different concepts. In the General Linear Model, we model the response as a linear combination that is linear in parameters (e.g. B0 + B1 + B2 etc.) which is different from assuming that there exists a linear relationship between some predictor x and the response y. The assumption of this linear relationship is a fundamental assumption of the method. 
A: This paragraph is a bit difficult to understand, so I'll give a small example to help unpack what it says in terms of assumptions. 
Imagine that y = body fat, x1 = body weight and x2 = body height and we are interested in investigating the relationship between y and x1 and x2 for all adult males in a specific community based on a random sample of 1000 males from that community.
The linear regression model relating y to x1 and x2 can then be stated as:
y = beta0 + beta1*x1 + beta2*x2 + error.  (*)

In the context of this example, the paragraph higlighted in green in your post should be interpreted as:


*

*y (body fat) is linearly related to x1 (body weight) for all the adult males in the community who share the same body height (that is, who have the same value for x2);

*y (body fat) is linearly related to x2 (body height) for all the adult males in the community who share the same body fat (that is, who have the same value for x1).


So the assumptions are that y is linearly related to x1 for each value of x2 and y is linearly related to x2 for each value of x1.  The assumption of linearity concerns two variables (e.g., y and x1) after controlling for the effect of the other predictor variable (e.g., x2). The parameter beta1 is just a way to quantify this relationship, as it indicates how fast y changes with x1 among adult males with the same value of x2. Similarly, the parameter beta2 indicates how fast y changes with x2 among adult males with the same value of x1. 
Usually we check these assumptions by plotting the residuals obtained after fitting model (*) to the data separately against each of x1 and x2. If those plots show no systematic pattern, then the assumptions are supported by the data. If one of the plots shows a systematic pattern (e.g., a quadratic pattern), that's an indication that the respective predictor has a nonlinear effect on y after controlling for the effect of the other predictor.
